Lines come up in rendering after adding hair
Try changing the dropdown menu in Hair BSDF from Reflection to Transmition. This helped when I downloaded the .blend file.

You could also try lowering the samples, as you said the problem only appeared when the count got up high. In the image rendering from the .blend file, noticable changes stop around 300-400 samples. More than 250, in general, just makes you feel better without actually changing anything.

Hope this helps and happy Blending!